“No sons of mine are dishonest men;
All honest men are men treated with respect”.
We can now construct our Dictionary, viz. m = honest; x = sons of mine; y = treated with respect.
(Note that the expression “x = sons of mine” is an abbreviated form of “x = the Differentia of ‘sons of mine’, when regarded as a Species of ‘men’”.)
The next thing is to translate the proposed Premisses into abstract form, as follows:—
“No x are m′;
All m are y”.
Next, by the process described at p. 50, we represent these on a Triliteral Diagram, thus:—
Next, by the process described at p. 53, we transfer to a Biliteral Diagram all the information we can.
The result we read as “No x are y′” or as “No y′ are x,” whichever we prefer. So we refer to our Dictionary, to see which will look best; and we choose
“No x are y′”,
which, translated into concrete form, is
“No son of mine fails to be treated with respect”.
(2)
“All cats understand French;
Some chickens are cats”.
Taking “creatures” as Univ., we write these as follows:—
“All cats are creatures understanding French;
Some chickens are cats”.
We can now construct our Dictionary, viz. m = cats; x = understanding French; y = chickens.
The proposed Premisses, translated into abstract form, are
“All m are x;
Some y are m”.
In order to represent these on a Triliteral Diagram, we break up the first into the two Propositions to which it is equivalent, and thus get the three Propositions
(1) “Some m are x;
(2) No m are x′;
(3) Some y are m”.
The Rule, given at p. 50, would make us take these in the order 2, 1, 3.
This, however, would produce the result
So it would be better to take them in the order 2, 3, 1. Nos. (2) and (3) give us the result here shown; and now we need not trouble about No. (1), as the Proposition “Some m are x” is already represented on the Diagram.
Transferring our information to a Biliteral Diagram, we get
This result we can read either as “Some x are y” or “Some y are x”.
After consulting our Dictionary, we choose
“Some y are x”,
which, translated into concrete form, is
“Some chickens understand French.”
(3)
“All diligent students are successful;
All ignorant students are unsuccessful”.
Let Univ. be “students”; m = successful; x = diligent; y = ignorant.
These Premisses, in abstract form, are
“All x are m;
All y are m′”.
These, broken up, give us the four Propositions
(1) “Some x are m;
(2) No x are m′;
(3) Some y are m′;
(4) No y are m”.
which we will take in the order 2, 4, 1, 3.
Representing these on a Triliteral Diagram, we get
And this information, transferred to a Biliteral Diagram, is
Here we get two Conclusions, viz.
“All x are y′;
All y are x′.”
And these, translated into concrete form, are
“All diligent students are (not-ignorant, i.e.) learned;
All ignorant students are (not-diligent, i.e.) idle”.
(4)
“Of the prisoners who were put on their trial at the last
Assizes, all, against whom the verdict ‘guilty’ was
returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also
sentenced to hard labour”.
Let Univ. be “the prisoners who were put on their trial at the last Assizes”; m = who were sentenced to imprisonment; x = against whom the verdict ‘guilty’ was returned; y = who were sentenced to hard labour.
The Premisses, translated into abstract form, are
“All x are m;
Some m are y”.
Breaking up the first, we get the three
(1) “Some x are m;
(2) No x are m′;
(3) Some m are y”.
Representing these, in the order 2, 1, 3, on a Triliteral Diagram, we get
Here we get no Conclusion at all.
You would very likely have guessed, if you had seen only the Premisses, that the Conclusion would be
“Some, against whom the verdict ‘guilty’ was returned,
were sentenced to hard labour”.
But this Conclusion is not even true, with regard to the Assizes I have here invented.
“Not true!” you exclaim. “Then who were they, who were sentenced to imprisonment and were also sentenced to hard labour? They must have had the verdict ‘guilty’ returned against them, or how could they be sentenced?”
Well, it happened like this, you see. They were three ruffians, who had committed highway-robbery. When they were put on their trial, they pleaded ‘guilty’. So no verdict was returned at all; and they were sentenced at once.]
I will now work out, in their briefest form, as models for the Reader to imitate in working examples, the above four concrete Problems.
(1)
“No son of mine is dishonest;
People always treat an honest man with respect.”
Univ. “men”; m = honest; x = my sons; y = treated with respect.
“No x are m′;
All m are y.”
∴ “No x are y′.”
i.e. “No son of mine ever fails to be treated with respect.”
(2)
“All cats understand French;
Some chickens are cats”.
Univ. “creatures”; m = cats; x = understanding French; y = chickens.
“All m are x;
Some y are m.”
∴ “Some y are x.”
i.e. “Some chickens understand French.”
(3)
“All diligent students are successful;
All ignorant students are unsuccessful”.
Univ. “students”; m = successful; x = diligent; y = ignorant.
“All x are m;
All y are m′.”
∴ “All x are y′;
All y are x′.”
i.e. “All diligent students are learned; and all ignorant students are idle”.
(4)
“Of the prisoners who were put on their trial at the last Assizes, all, against whom the verdict ‘guilty’ was returned, were sentenced to imprisonment;
Some, who were sentenced to imprisonment, were also sentenced to hard labour”.
Univ. “prisoners who were put on their trial at the last Assizes”, m = sentenced to imprisonment; x = against whom the verdict ‘guilty’ was returned; y = sentenced to hard labour.
“All x are m;
Some m are y.”
There is no Conclusion.
[Review Tables VII, VIII (pp. 48, 49). Work Examples § 1, 17–21 (p. 97); § 4, 1–6 (p. 100); § 5, 1–6 (p. 101).]
§ 3.
Given a Trio of Propositions of Relation, of which every two contain a Pair of codivisional Classes, and which are proposed as a Syllogism; to ascertain whether the proposed Conclusion is consequent from the proposed Premisses, and, if so, whether it is complete.
The Rules, for doing this, are as follows:—
(1) Take the proposed Premisses, and ascertain, by the process described at p. 60, what Conclusion, if any, is consequent from them.
(2) If there be no Conclusion, say so.
(3) If there be a Conclusion, compare it with the proposed Conclusion, and pronounce accordingly.
I will now work out, in their briefest form, as models for the Reader to imitate in working examples, six Problems.
(
1)
“All soldiers are strong;
All soldiers are brave.
Some strong men are brave.”
Univ. “men”; m = soldiers; x = strong; y = brave.
“All m are x;
All m are y.
Some x are y.”
∴ “Some x are y.”
Hence proposed Conclusion is right.
(2)
“I admire these pictures;
When I admire anything I wish to examine it thoroughly.
I wish to examine some of these pictures thoroughly.”
Univ. “things”; m = admired by me; x = these pictures; y = things which I wish to examine thoroughly.
“All x are m;
All m are y.
Some x are y.”
∴ “All x are y.”
Hence proposed Conclusion is incomplete, the complete one being “I wish to examine all these pictures thoroughly”.
(3)
“None but the brave deserve the fair;
Some braggarts are cowards.
Some braggarts do not deserve the fair.”
Univ. “persons”; m = brave; x = deserving of the fair; y = braggarts.
“No m′ are x;
Some y are m′.
Some y are x′.”
∴ “Some y are x′.”
Hence proposed Conclusion is right.
(4)
“All soldiers can march;
Some babies are not soldiers.
Some babies cannot march”.
Univ. “persons”; m = soldiers; x = able to march; y = babies.
“All m are x;
Some y are m′.
Some y are x′.”
There is no Conclusion.
(5)
“All selfish men are unpopular;
All obliging men are popular.
All obliging men are unselfish”.
Univ. “men”; m = popular; x = selfish; y = obliging.
“All x are m′;
All y are m.
All y are x′.”
∴ “All x are y′;
All y are x′.”
Hence proposed Conclusion is incomplete, the complete one containing, in addition, “All selfish men are disobliging”.
(6)
”No one, who means to go by the train and cannot get a conveyance, and has not enough time to walk to the station, can do without running;
This party of tourists mean to go by the train and cannot get a conveyance, but they have plenty of time to walk to the station.
This party of tourists need not run.”
Univ. “persons meaning to go by the train, and unable to get a conveyance”; m = having enough time to walk to the station; x = needing to run; y = these tourists.
“No m′ are x′;
All y are m.
All y are x′.”
There is no Conclusion.
[Here is another opportunity, gentle Reader, for playing a trick on your innocent friend. Put the proposed Syllogism before him, and ask him what he thinks of the Conclusion.
He will reply “Why, it’s perfectly correct, of course! And if your precious Logic-book tells you it isn’t, don’t believe it! You don’t mean to tell me those tourists need to run? If I were one of them, and knew the Premisses to be true, I should be quite clear that I needn’t run—and I should walk!”
And you will reply “But suppose there was a mad bull behind you?”
And then your innocent friend will say “Hum! Ha! I must think that over a bit!”
You may then explain to him, as a convenient test of the soundness of a Syllogism, that, if circumstances can be invented which, without interfering with the truth of the Premisses, would make the Conclusion false, the Syllogism must be unsound.]
[Review Tables V–VIII (pp. 46–49). Work Examples § 4, 7–12 (p. 100); § 5, 7–12 (p. 101); § 6, 1–10 (p. 106); § 7, 1–6 (pp. 107, 108).]
BOOK VI.
THE METHOD OF SUBSCRIPTS.
CHAPTER I.
INTRODUCTORY.
Let us agree that “x1” shall mean “Some existing Things have the Attribute x”, i.e. (more briefly) “Some x exist”; also that “xy1” shall mean “Some xy exist”, and so on. Such a Proposition may be called an ‘Entity.’
[Note that, when there are two letters in the expression, it does not in the least matter which stands first: “xy1” and “yx1” mean exactly the same.]
Also that “x0” shall mean “No existing Things have the Attribute x”, i.e. (more briefly) “No x exist”; also that “xy0” shall mean “No xy exist”, and so on. Such a Proposition may be called a ‘Nullity’.
Also that “†” shall mean “and”.
[Thus “ab1 † cd0” means “Some ab exist and no cd exist”.]
Also that “¶” shall mean “would, if true, prove”.
[Thus, “x0 ¶ xy0” means “The Proposition ‘No x exist’ would, if true, prove the Proposition ‘No xy exist’”.]
When two Letters are both of them accented, or both not accented, they are said to have ‘Like Signs’, or to be ‘Like’: when one is accented, and the other not, they are said to have ‘Unlike Signs’, or to be ‘Unlike’.
CHAPTER II.
REPRESENTATION OF PROPOSITIONS OF RELATION.
Let us take, first, the Proposition “Some x are y”.
This, we know, is equivalent to the Proposition of Existence “Some xy exist”. Hence it may be represented by the expression “xy1”.
The Converse Proposition “Some y are x” may of course be represented by the same expression, viz. “xy1”.
Similarly we may represent the three similar Pairs of Converse Propositions, viz.—
“Some x are y′” = “Some y′ are x”,
“Some x′ are y” = “Some y are x′”,
“Some x′ are y′” = “Some y′ are x′”.
Let us take, next, the Proposition “No x are y”.
This, we know, is equivalent to the Proposition of Existence “No xy exist”. Hence it may be represented by the expression “xy0”.
The Converse Proposition “No y are x” may of course be represented by the same expression, viz. “xy0”.
Similarly we may represent the three similar Pairs of Converse Propositions, viz.—
“No x are y′” = “No y′ are x”,
“No x′ are y” = “No y are x′”,
“No x′ are y′” = “No y′ are x′”.
Let us take, next, the Proposition “All x are y”.
Now it is evident that the Double Proposition of Existence “Some x exist and no xy′ exist” tells us that some x-Things exist, but that none of them have the Attribute y′: that is, it tells us that all of them have the Attribute y: that is, it tells us that “All x are y”.
Also it is evident that the expression “x1 † xy′0” represents this Double Proposition.
Hence it also represents the Proposition “All x are y”.
[The Reader will perhaps be puzzled by the statement that the Proposition “All x are y” is equivalent to the Double Proposition “Some x exist and no xy′ exist,” remembering that it was stated, at p. 33, to be equivalent to the Double Proposition “Some x are y and no x are y′” (i.e. “Some xy exist and no xy′ exist”). The explanation is that the Proposition “Some xy exist” contains superfluous information. “Some x exist” is enough for our purpose.]
This expression may be written in a shorter form, viz. “x1y′0”, since each Subscript takes effect back to the beginning of the expression.
Similarly we may represent the seven similar Propositions “All x are y′”, “All x′ are y”, “All x′ are y′”, “All y are x”, “All y are x′”, “All y′ are x”, and “All y′ are x′”.
[The Reader should make out all these for himself.]
It will be convenient to remember that, in translating a Proposition, beginning with “All”, from abstract form into subscript form, or vice versâ, the Predica
te changes sign (that is, changes from positive to negative, or else from negative to positive).
[Thus, the Proposition “All y are x′” becomes “y1x0”, where the Predicate changes from x′ to x.
Again, the expression “x′1y′0” becomes “All x′ are y”, where the Predicate changes for y′ to y.]
CHAPTER III.
SYLLOGISMS.
§ 1.
Representation of Syllogisms.
We already know how to represent each of the three Propositions of a Syllogism in subscript form. When that is done, all we need, besides, is to write the three expressions in a row, with “†” between the Premisses, and “¶” before the Conclusion.
[Thus the Syllogism
“No x are m′;
All m are y.
∴ No x are y′.”
may be represented thus:—
xm′0 † m1y′0 ¶ xy′0
When a Proposition has to be translated from concrete form into subscript form, the Reader will find it convenient, just at first, to translate it into abstract form, and thence into subscript form. But, after a little practice, he will find it quite easy to go straight from concrete form to subscript form.]
§ 2.
Formulæ for solving Problems in Syllogisms.
When once we have found, by Diagrams, the Conclusion to a given Pair of Premisses, and have represented the Syllogism in subscript form, we have a Formula, by which we can at once find, without having to use Diagrams again, the Conclusion to any other Pair of Premisses having the same subscript forms.
[Thus, the expression
xm0 † ym′0 ¶ xy0
is a Formula, by which we can find the Conclusion to any Pair of Premisses whose subscript forms are
xm0 † ym′0
For example, suppose we had the Pair of Propositions
Complete Works of Lewis Carroll Page 104